Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 715

that even viaεi,tboth short- and long-run dependence can be introduced into the

panel. The same applies for the vectorut=

(

u1,t,...,uN,t

)′
.
In a way, short-run dependence can, in several respects be potentially less prob-
lematic than long-run dependence. If we think of, for example, unit root testing,
then short-run cross-sectional correlation will lead to distorted inference, of which
account can be taken. Compare the discussion of O’Connell (1998) or the set-up of
Chang (2002). Long-run dependence typically is found to have more detrimental
effects, see Lyhagen (2000), that cannot be remedied easily, for example, by simple
feasible GLS-type corrections.
There is one further issue with respect to long-run dependence or cross-unit coin-
tegration. Our discussion up to here has been for the case of fixed cross-sectional
dimension. In general, the dimension of the cross-unit cointegrating space is
itself dependent uponNsince, due to the inclusion of additional panel mem-
bers, additional cross-unit cointegrating relationships may emerge. Consequently,
a thorough analysis of cross-unit dependence and its effects needs to consider the
dependence behavior whenN→∞. Part of the literature takes short-cuts in this
respect; for example, Evans and Karras (1996) and Lyhagen (2000) consider a panel
set-up with exactly one common trend and with all pair-wise differences of the
series being stationary for all values ofN. In this way they avoid a proper con-
sideration of the dependence of the cointegration structure on the cross-sectional
dimension.
Considering all series together as a high-, or in the limit infinitely-, dimensional
time series is useful for understanding the algebraic structure of cointegration,
cross-unit cointegration and short-run cross-sectional dependence. However, from
a panel modeling perspective, restrictions on the joint DGP have to be put in place
in order to materialize gains from pooling in one way or another.

Looking at factor models, consider the joint processut=

(

u1,t,...,uN,t

)′

, where

for simplicity we ignore deterministic components:

ut=′Ft+et,

with=

[

π

′

1 ,...,π

′

N

]′

∈Rr×N, thercommon factorsFt∈Rr, and the idiosyncratic

componentset=

(

e1,t,...,eN,t

)′
.
Since we focus here on the cointegration implications of the factor model, we
assume for simplicity that the factor loadings matrixis non-stochastic and that
the idiosyncratic components are cross-sectionally independent. As noted pre-
viously, the fact that Bai and Ng (2004) also allow the factor loadingsto be
stochastic is mainly a mathematical achievement but does not really change any of
the properties of the time series panel since all observations are generated fromone
single realizationof the factor loadings. Bai and Ng (2004) allow for a certain form
of correlation between the components ofet(described in section 13.2.2.2) which
is why they dub their model an approximate factor model. The corresponding
assumption (see Bai and Ng, 2004, p. 1130, Assumption C) is, however, of mainly
a technical character and simply allows for bounded correlation between the